I. I. Shapiro

Viking relativity experiment: Verification of signal retardation by solar gravity

Analysis of 14 months of data obtained from radio ranging to the Viking spacecraft verified, to an estimated accuracy of 0.1%, the prediction of the general theory of relativity that the round-trip times of light signals traveling between the earth and Mars are increased by the direct effect of solar gravity. The corresponding value for the metric parameter gamma is 1.000 plus or minus 0.002, where the quoted uncertainty, twice the formal standard deviation, allows for possible systematic errors.

Forced nutations of the Earth: Influence of inner core dynamics: 1. Theory

Gravitational and pressure couplings between the solid inner core and the rest of the Earth give rise to torques through which the inner core influences the nutational motions of the Earth. In view of the very small magnitude of the moment of inertia of the inner core relative to that of the the Earth as a whole, one expects from physical considerations that inclusion of the inner core in the dynamics should lead to a new nutational normal mode with a natural frequency not too far from that of the free core nutation, and to an associated weak resonance in the amplitude of forced nutations. We present here a treatment of the nutation problem for an oceanless, elastic, spheroidally stratified Earth, with the dynamical role of the inner core explicitly included in the formulation. As a preliminary to the setting-up of dynamical equations, we devote some attention to a careful definition of a suitable coordinate system and of certain basic dynamical variables. We use the approach of Sasao et al. (1980), with their system of dynamical equations enlarged by the inclusion of two additional equations which are needed to describe the rotational motion of the inner core. An extension and sharpening of a line of reasoning employed by them enables us to derive expressions for the torques which couple the mantle and the fluid outer core to the solid inner core. Solving the enlarged system of equations, we show that a new nearly diurnal eigenfrequency does emerge; a rough estimate places it not very far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Another eigenfrequency, attributable to a wobble of the inner core, is also found; its value is estimated to be a few times smaller than the wobble frequency that the inner core would have in the absence of couplings to the rest of the Earth. Considering an expansion, in terms of resonance contributions, of the amplitude of forced nutations normalized relative to that for a corresponding rigid Earth model, we indicate how the coefficients in the expansion are related to those in expansions of the type used by Wahr (1981b). Finally, we discuss the problem of comparing observed nutation amplitudes with those computed on the basis of Earth models generated from seismological data, with special reference to the fact that the dynamical ellipticity of the Earth, as computed from published Earth models which assume the condition of hydrostatic equilibrium, differs significantly from that determined from the precession constant. Numerical results, corrections for unmodeled effects, and comparison with observational results will be dealt with in accompanying papers.

Geodesy by radio interferometry: Water vapor radiometry for estimation of the wet delay

An important source of error in very-long-baseline interferometry (VLBI) estimates of baseline length is unmodeled variations of the refractivity of the neutral atmosphere along the propagation path of the radio signals. We present and discuss the method of using data from a water vapor radiometer (WVR) to correct for the propagation delay caused by atmospheric water vapor, the major cause of these variations. Data from different WVRs are compared with estimated propagation delays obtained by Kalman filtering of the VLBI data themselves. The consequences of using either WVR data or Kalman filtering to correct for atmospheric propagation delay at the Onsala VLBI site are investigated by studying the repeatability of estimated baseline lengths from Onsala to several other sites. The lengths of the baselines range from 919 to 7941 km. The repeatability obtained for baseline length estimates shows that the methods of water vapor radiometry and Kalman filtering offer comparable accuracies when applied to VLBI observations obtained in the climate of the Swedish west coast. For the most frequently measured baseline in this study, the use of WVR data yielded a 13% smaller weighted-root-mean-square (WRMS) scatter of the baseline length estimates compared to the use of a Kalman filter. It is also clear that the “best” minimum elevation angle for VLBI observations depends on the accuracy of the determinations of the total propagation delay to be used, since the error in this delay increases with increasing air mass. For use of WVR data along with accurate determinations of total surface pressure, the best minimum is about 20 degrees; for use of a model for the wet delay based on the humidity and temperature at the ground, the best minimum is about 35 degrees.

2237 + 0305: A new and unusual gravitational lens

A new gravitational lens system has been discovered which consists of a quasar at z = 1.695 nearly centered on a 15-mag spiral galaxy, 2237 + 0305, at z = 0.0394. At 2 arcsec resolution, only a single optical image of the quasar is visible; its centroid is located within approximately 0.3 arcsec of the center of the galaxy. Snapshot observations at the VLA yielded no detectable radio radiation, placing an upper limit of about 0.5 mJy on the combined flux density of the galaxy and quasar at a wavelength of 6 cm. A simple gravitational lens model that accounts for these observations is discussed. 17 references.

Geodesy using the Global Positioning System: The effects of signal scattering on estimates of site position

Analysis of Global Positioning System (GPS) data from two sites separated by a horizontal distance of only ∼2.2 m yielded phase residuals exhibiting a systematic elevation angle dependence. One of the two GPS antennas was mounted on an ∼1-m-high concrete pillar, and the other was mounted on a standard wooden tripod. We performed elevation angle cutoff tests with these data and established that the estimate of the vertical coordinate of site position was sensitive to the minimum elevation angle (elevation cutoff) of the data analyzed. For example, the estimate of the vertical coordinate of site position changed by 9.7±0.8 mm when the minimum elevation angle was increased from 10° to 25°. We performed simulations based on a simple (ray tracing) multipath model with a single horizontal reflector which demonstrated that the results from the elevation angle cutoff tests and the pattern of the residuals versus elevation angle could be qualitatively reproduced if the reflector were located 0.1–0.2 m beneath the antenna phase center. We therefore hypothesized that the elevation-angle-dependent error was caused by scattering from the horizontal surface of the pillar, located a distance of ∼0.2 m beneath the antenna phase center. We tested this hypothesis by placing microwave absorbing material between the antenna and the pillar in a number of configurations and by analyzing the changes in apparent position of the antenna. The results indicate that (1) the horizontal surface of the pillar is indeed the main scatterer, (2) both the concrete and the metal plate embedded in the pillar are significant sources of scattering, and (3) the scattering can be reduced greatly by the use of microwave absorbing materials. These results have significant implications for the accuracy of global GPS geodetic tracking networks which use pillar-antenna configurations identical or similar to the one used for this study at the Westford WFRD GPS site.

Forced nutations of the Earth: Influence of inner core dynamics: 2. Numerical results and comparisons

We apply the theory developed in Paper 1 (Mathews et al., this issue), which includes the solid inner core explicitly in the dynamical equations, to obtain the eigenfrequencies and other characteristics of the Earths nutational normal modes as well as the amplitudes of forced nutations at various tidal frequencies, for two commonly used Earth models, 1066A and the Preliminary Reference Earth Model (PREM). We also make an evaluation of various procedures for taking account of known deviations of the Earth from models, notably in the dynamical ellipticity e for which the two models yield values which are over 1% smaller than the value e* deduced from the precession constant. On adopting the procedure of simply replacing e by e* in the equations of our theory, the values obtained for some of the nutation amplitudes for model 1066A differ significantly from the corresponding results of Wahr (1981b). The largest of the differences, which occur in the prograde semiannual, retrograde 18.6 year, and retrograde annual nutation terms, amount to −0.59, 0.35, and −0.25 milliarcseconds (mas), respectively, while the standard errors in the very long baseline interferometry (VLBI) determinations are now only about 0.04 mas except in the long period terms. The difference in the procedures used to take account of the discrepancy between e and e* contributes −0.56, 0.81, and −0.17 mas, respectively, to the above-noted differences. For the purpose of comparison with VLBI-observed data, we use the results for a “modified PREM,” defined by a set of Earth model parameters which differ from those of PREM only in having e* for the dynamical ellipticity of the Earth as a whole and a modified value for the dynamical ellipticity eƒ of the fluid outer core. The amplitudes computed for this model, with corrections applied for the effects of ocean tides and mantle anelasticity, are in generally satisfactory agreement with observed values, when the modified eƒ is determined by matching the theoretical and observed values for the retrograde annual term. (The modified eƒ is 0.002665, about 4.6% higher than in PREM, equivalent to an increase, relative to PREM, of about 430 meters in the difference between the equatorial and the polar radii of the core-mantle boundary. We find that contributions from inner-core dynamics to the prograde semiannual and annual, and the retrograde 18.6 year and annual terms, recomputed for modified PREM, amount to −0.09, 0.03, −0.36, and −0.09 mas, respectively.) The largest residual remaining, other than in the long-period terms which still have an uncertainty of about 1 mas, is −0.25 mas in the prograde fortnightly amplitude. Consideration of possible sources of the discrepancies is facilitated by a resonance expansion of the amplitude of forced nutations, as a function of frequency, normalized relative to that for a rigid Earth model. We also provide tables which exhibit the sensitivities of various relevant quantities (the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as the nutation amplitudes at important tidal frequencies) to possible errors in the Earth parameters which enter our theory. Reconciliation of theoretical and experimental values for the prograde fortnightly term, for instance, could be accomplished, without affecting significantly the comparison for other nutation terms, by a decrease of about 10% in the value of the compliance parameter k that represents, in effect, the deformability of the Earth as a whole in response to perturbations of its rotation; but this change in k would have to be produced by some mechanism which does not affect the values of the other compliances.

A spectral formalism for computing three-dimensional deformations due to surface loads. 2: Present-day glacial isostatic adjustment

Using a spherically symmetric, self-gravitating, linear viscoelastic Earth model, we predict present-day three-dimensional surface deformation rates and baseline evolutions arising as a consequence of the late Pleistocene glacial cycles. In general, we use realistic models for the space-time geometry of the final late Pleistocene deglaciation event and incorporate a gravitationally self-consistent ocean meltwater redistribution. The predictions of horizontal velocity presented herein differ significantly, in both their amplitude and their spatial variation, from those presented in an earlier analysis of others which adopted simplified models of both the late Pleistocene ice history and the Earth rheology. An important characteristic of our predicted velocity fields is that the melting of the Laurentide ice sheet over Canada is capable of contributing appreciably to the adjustment in Europe. The sensitivity of the predictions to variations in mantle rheology is investigated by considering a number of different Earth models, and by computing appropriate Frechet kernels. These calculations suggest that the sensitivity of the deformations to the Earths rheology is significant and strongly dependent on the location of the site relative to the ancient ice sheet. The effects on the predictions of three-dimensional deformation rates of altering the ice history or adopting approximate models for the ocean meltwater redistribution have also been considered and found to be important (the former especially so). Finally, for a suite of Earth models we provide predictions of the velocity of a number of baselines in North America and Europe. We find that, in general, both radial and tangential motions contribute significantly to baseline length changes, and that these contributions are a strong function of the Earth model. We have, furthermore, found a set of Earth models which, together with the ICE-3G deglaciation chronology, produce predictions of baseline length changes that are consistent with very long baseline interferometry measurements of baselines within Europe.

A spectral formalism for computing three‐dimensional deformations due to surface loads: 1. Theory

We outline a complete spectral formalism for computing high spatial resolution three-dimensional deformations arising from the surface mass loading of a spherically symmetric planet. The main advantages of the formalism are that all surface mass loads are always described using a consistent mathematical representation (expansions using spherical harmonic basis functions) and that calculations of deformation fields for various spatial resolutions can be performed by simply altering the spherical harmonic degree truncation level of the procedure. The latter may be important when incorporating improved observational constraints on a particular surface mass load, when considering potential errors in the computed field associated with mass loading having a spatial scale unresolved by the observational constraints, or when treating a number of global surface mass loads constrained with different spatial resolutions. These advantages do not extend to traditional “Greens function” approaches which involve surface element discretizations of the global mass loads. In the practical application of the Greens function approach the discretization is subjective and dependent on the particular surface mass load being considered. Furthermore, treatment of mass loads with higher spatial resolutions can require tedious rediscretization of the surface elements. Another advantage of the spectral formalism, over the Greens function approach, is that a posteriori analyses of the computed deformation fields, such as degree correlations, power spectra, and filtering analyses, are easily performed. In developing the spectral formalism, we consider specific cases where the Earths mantle is assumed to respond as an elastic, slightly anelastic, or linear viscoelastic medium. In the case of an elastic or slightly anelastic mantle rheology the spectral response equations incorporate frequency dependent Love numbers. The formalism can therefore be used, for example, to compute the potentially resonant deformational response associated with the free core nutation and Chandler wobble eigenfunctions. For completeness, the spectral response equations include both body forces, as arise from the gravitational attraction of the Sun and the Moon, and surface mass loads. In either case, and for both elastic and anelastic mantle rheologies, we outline a pseudo-spectral technique for computing the ocean adjustment associated with the total gravitational perturbation induced by the external forcing. Three-dimensional deformations computed using the usual Love number approach are generally referenced to an origin located at the center of mass of the undeformed planet. We derive a spectral technique for transforming the results to an origin located at the center of mass of the deformed planet.

Forced nutations of the Earth: influence of inner core dynamics 3. Very long interferometry data analysis

We discuss the analysis of 798 very long baseline interferometry (VLBI) experiments carded out between July 1980 and February 1989, and the determination from this analysis of corrections to selected coefficients in the International Astronomical Union (IAU) 1980 theory of the nutations of the Earth. Our analysis confirms earlier VLBI results and indicates that most of these corrections can be explained by carefully accounting for (1) corrections to the IAU 1980 rigid-Earth nutation series, (2) the presence of the Earths inner core, (3) the difference between the dynamic flattening of the Earth inferred from the precession constant and that inferred from seismic models of the internal density structure of the Earth, and (4) the effects of mantle anelasticity and ocean tides. The standard deviations of the corrections to the coefficients are 0.04 milliarcseconds (mas) for terms with periods under 430 days, and 1.0 mas for the terms with a period of 18.6 years. The unresolved issues raised by our analysis are the origins of corrections to the out-of-phase retrograde annual(0.39 mas) and the in-phase prograde 13.66 day (− 0.25 mas) nutations. Our analysis also yields a correction to the IAU 1976 value for the luni-solar precession constant of −0.32±0.13 arc sec/century (cy).

Precision Geodesy Using the Mark-III Very-Long-Baseline Interferometer System

Very-long-baseline interferometry (VLBI) has been used to make precise measurements of the vector separation between widely separated antennas. The system for acquiring and processing VLBI data known as Mark-III is described. Tests of the system show it to have millimeter-level accuracy on short baselines; measurements of baselines longer than a few hundred kilometers suggest that accuracy is limited by the uncertainty in the calibration of tropospheric path delay to the level of a few centimeters. VLBI experiments conducted between 1976 and 1983 have demonstrated the stability of the North American plate by showing that there is no change in the distance between easternl-California and Massachusetts at the level of a few millimeters per year or greater. Experiments made from 1980 to 1984 indicate that the distance from Massachusetts to Sweden is increasing by 1.7 ± 1 cm/year where the quoted standard deviation includes the estimated effects of systematic atic errors